Why the set of irrational numbers is represented as $\mathbb{R}\setminus\mathbb{Q}$ instead of $\mathbb{R}-\mathbb{Q}$? What does the "\" symbol means in this context?
I have seen it used for quotient sets like $X /{\sim}$ where $X$ is a set and $\sim$ is an equivalence relation but I don't know what it means applied to two sets.
$\mathbb{R}-\mathbb{Q}$ seems to be much more suitable, since the set of irrational numbers are just that: real numbers which are not rational.
 A: Both symbols $\setminus$ \setminus and $-$ - are used for denoting set difference: $$A\setminus B = A - B = \{ x \mid x \in A,\,x \not\in B  \}.$$
I, particularly, prefer $A \setminus B$. In some contexts, we can have something like: $$A-B = \{ x-y \mid x \in A,\, y \in B  \},$$ so sticking to $\setminus$ there is zero chance of confusion.
A: No answer on this page has pointed out that $\setminus$ is used like $-$ for set subtraction very often in math, but that this is very different from $/$. In particular $/$ is used for quotients, which are a completely different thing.
A: $\setminus$ and $-$ mean the exact same thing in set theory, both mean set difference. 
A: The ordinary minus "$-$" and the ordinary slash "$/$" both have various meanings when used with sets, depending on context. The backslash "$\setminus$" is used exclusively for relative complement (i.e., set subtraction) as far as I have ever seen.
A: Both are used to represent the set of irrational numbers 
see this link
They are the same exact thing. 
$a \in \mathbb{R \backslash Q}$  means that $a$ is a real number that is not a quotient which can also be denoted as $a \in \mathbb{R - Q}$
